In this paper we illustrate the relationship between defects and symmetries for a simple
class of symmetries of two-dimensional Yang-Mills theory. Yang-Mills
theory on a Riemann surface has a long and rich history as an exactly
solvable quantum gauge theory which is the first example of a
non-abelian gauge theory that can be reformulated
as a (topological) string theory (see [1]} for a
review). Mathematically it has served as a tool for studying the
topology of various moduli spaces of interest in geometry and
dynamical systems, such as the
moduli spaces of flat connections [2]}, [3]}, [4]}, the Hurwitz moduli spaces of branched
coverings [1]}, [6]}, and the
principal moduli spaces of holomorphic
differentials [7]}, [8]}. In the following
we will study how some of these features are modified in the
presence of domain walls, which in two dimensions can be thought of as
symmetry twist branch cuts on the surface [9]}, corresponding to outer automorphisms of
the gauge group.Similar defects are constructed
in [10]} in the context of
three-dimensional Dijkgraaf-Witten theories, and have also been
proposed to have physical realisations in certain topological states
of matter, see for example [11]}.
